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Naturally reductive Killing $$f$$-manifolds. (English. Russian original) Zbl 0977.53025
Russ. Math. Surv. 54, No. 3, 623-625 (1999); translation from Usp. Mat. Nauk 54, No. 3, 151-152 (1999).
From the text: Amongst the most remarkable almost Hermitian structures $$(g,J)$$ on smooth manifolds $$M$$ are the nearly Kähler structures. The defining condition is the requirement that $$\nabla_X(J)X=0$$, where $$\nabla$$ is the Levi-Civita connection of the (pseudo) Riemannian metric $$g$$ and $$X$$ is an arbitrary smooth vector field on $$M$$. The Killing $$f$$-structures are a generalization of this concept to metric $$f$$-structures of classical type (those with $$f^3+f=0)$$, and the defining relation here is $$\nabla_X (f)X=0$$. In this article we analyse the possible existence of invariant Killing $$f$$-structures on naturally reductive homogeneous spaces. In particular, criteria are obtained for canonical $$f$$-structures on homogeneous $$\Phi$$-spaces of orders 4 and 5 to be Killing.

MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C30 Differential geometry of homogeneous manifolds
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